# Subschemes of projective varieties

I'm studying the article Introduction to Lawson Homology of Peters and Kosarew. In their exposition of Lawson homology they call projective variety any zero-locus $X$ of homogeneous polynomials in the complex projective space and, in particular, they consider Chow varieties $\mathcal{C}_{p,d}(X)$ only for varieties of this type.

At page 8 they give the following definition-lemma.

Then they say:

From this one see that $Z\in \mathcal{C}_{m+t,d}(T\times X)$ and this leads to my question:

Question: Can one see a subscheme of a projective variety as an algebraic cycle?

If $$X$$ is a Noetherian scheme, there is a map \begin{align*} \{\text{subschemes of X}\} &\to \operatorname{CH}_*(X)\\ Z &\mapsto [Z], \end{align*} defined by $$[Z] = \sum_{i=1}^r m_i [Z_i],$$ where the $$Z_i \subseteq Z$$ are the (reduced) irreducible components of $$Z$$, and $$m_i = \ell_{\mathcal O_{Z,\eta_i}}(\mathcal O_{Z,\eta_i})$$ is the length of the scheme $$Z$$ at the generic point $$\eta_i$$ of $$Z_i$$.

Example. If $$Z$$ is reduced, then $$m_i = 1$$ for all $$i$$. The converse is not true; for example the subscheme $$Z = \operatorname{Spec} k[x,y]/(x^2,xy) \subseteq \mathbb A^2_k$$ is not reduced, but $$[Z] = [V(x)]$$ is the $$y$$-axis with multiplicity one. This example also shows that the map $$Z \mapsto [Z]$$ is not injective.

It is injective on reduced subschemes when viewed as a map to $$Z_*(X)$$: any reduced subscheme $$Z \subseteq X$$ is uniquely determined by the collection $$\{Z_i\}$$ of components of $$Z$$. But on Chow groups this is again false; for example any two lines in $$\mathbf P^2$$ have the same cycle class.

• Many thanks for the answer and the reference. Anyway, in the setup the authors of the paper are working with the Chow variety $C_{p,d}(X)$ of the projective variety $X$ is (set-theoretically) the free abelian monoid generated by irreducibles projective subvarieties of $X$. Now, they take a subscheme of the projective variety $X\times T$ and write $Z \in \mathcal{C}_{m+t,d}(T\times X)$ then I'm forced to think that there is a way to see $Z$ as $\sum_in_iV_i$ where the $V_i$projective irreducuble subvarieties of $X\times T$. I hope that this comment clarifies what I'm trying to understand, Aug 7, 2018 at 22:12
• I don't know all that much about Chow varieties. There is a chance that something different is going on that what I'm suggesting. For example, there is also a map $Z \mapsto [\mathcal O_Z]$, which has slightly different properties. Perhaps this version is closer to being injective and hence allowing you to think of subschemes as cycles. Aug 7, 2018 at 22:44